Prove the following theorem: Let R be any ring and let f =! 0 and g =! 0 (they don't equal zero) be polynomials in R[x]. If the leading coefficient of either f or g is a unit in R, then:
1) fg =! 0 in R[x]
2) deg(fg) = deg(f) + deg(g)
The Attempt at a Solution
So i set up two polynomials f and g, where |g| = n and |f| = m. I multiplied these together and showed that the last term is equal to x^(mn) multiplied by the summation of dual combinations of coefficients such that i+j = mn where i represents the i'th coefficient on in f and j represents the j-th coefficient in g. I'm trying to show that this summation does not equal zero, and therefore deg(fg) = mn = deg(f) + deg(g), which would also show that fg =! 0 in R[x]
Anyway, that's how far I am. I'm trying to figure out what the lead coefficient of f being a unit has to do with it... any advice would be koo koo kachu.